Bateman-Horn constants of monic quadratic polynomials

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Numbers

$f$

$C(f)$

x^2-30:

0.8601147104299838789007494560328252807806904261844759019415598420694685969143298397526824924379622660

comment: The discriminant is $\Delta=120$.

x^2-29:

1.126772839598820996273857212748371200099365764891258533100946802507352106104383260933955097581403555

comment: The discriminant is $\Delta=116$.

x^2-28:

0.7573712324586387280719344925230232384822718784416289567546386994027326451841426894862906511000715422

comment: The discriminant is $\Delta=112$.

x^2-27:

1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473

comment: The discriminant is $\Delta=108$.

x^2-26:

1.168725054064629519927263619162420026682081108668887085636491388421635766085867648737434684009878222

comment: The discriminant is $\Delta=104$.

x^2-24:

1.035755877734863854678232494176401896040827459259040307383421447325501006449749301048770496936346249

comment: The discriminant is $\Delta=96$.

x^2-23:

1.389165729225091021008641394438937900491782100374239392159940634714728765171873881196703677828836074

comment: The discriminant is $\Delta=92$.

x^2-22:

0.5977866238870616809009077305745767918130060035608807497940575013180175697794438719845945910594857504

comment: The discriminant is $\Delta=88$.

x^2-21:

0.9278845824259630964861938618708887690023142404813744256456850301459903522338283872702898889257291492

comment: The discriminant is $\Delta=84$.

x^2-20:

1.773305066089907147323629883821673371188039518505172125291641380792830065421848013626651820657820539

comment: The discriminant is $\Delta=80$.

x^2-19:

0.5442381158545459834694417829112003690997867548503895850374429765275947307063046967002063104561103165

comment: The discriminant is $\Delta=76$.

x^2-18:

1.233369606273401136389080676108783664530189797828033202904168142133405692834916133351665887818675312

comment: The discriminant is $\Delta=72$.

x^2-17:

2.360474794680180004013217069970078663275795672137371417024644858094859086906230212900083312385861010

comment: The discriminant is $\Delta=68$.

x^2-15:

0.9117194099555348423618460572171063721199634074407231727580630198006585349049350488326141429725867783

comment: The discriminant is $\Delta=60$.

x^2-14:

1.151675729617594721217200488222297587405377081685228924691542287580788442705518760485501579275648485

comment: The discriminant is $\Delta=56$.

x^2-13:

0.8072362275555245670688831014666742044407393694788056234907708193950660811802836289826826609765204381

comment: The discriminant is $\Delta=52$.

x^2-12:

1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473

comment: The discriminant is $\Delta=48$.

x^2-11:

1.147979957061047562456689725537627701501502268797270468461706803083155724182282459186860059680031253

comment: The discriminant is $\Delta=44$.

x^2-10:

0.6711139154474606895516481167249919229589063161426530994908947791974688362715536026551029106420576270

comment: The discriminant is $\Delta=40$.

x^2-8:

1.850054409410101704583621014163175496795284696742049804356252213200108539252374200027498831728012968

comment: The discriminant is $\Delta=32$.

x^2-7:

0.7573712324586387280719344925230232384822718784416289567546386994027326451841426894862906511000715422

comment: The discriminant is $\Delta=28$.

x^2-6:

1.035755877734863854678232494176401896040827459259040307383421447325501006449749301048770496936346249

comment: The discriminant is $\Delta=24$.

x^2-5:

1.773305066089907147323629883821673371188039518505172125291641380792830065421848013626651820657820539

comment: The discriminant is $\Delta=20$.

x^2-3:

1.383424289287262448110154553079870540545674213701469986295044165358293142966169406450557700560887473

comment: The discriminant is $\Delta=12$.

x^2-2:

1.850054409410101704583621014163175496795284696742049804356252213200108539252374200027498831728012968

comment: The discriminant is $\Delta=8$.

x^2+1:

1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984

comment: The discriminant is $\Delta=-4$; primes of the form $n^2+1$ are Landau's fourth problem [4]; OEIS A199401 gives this normalization, A331941 gives half of it, and A206709 counts such primes [8].

x^2+2:

0.7130631042401929902232626073998268375684482408392256661074012026495749206783764001179141688701402792

comment: The discriminant is $\Delta=-8$.

x^2+3:

1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868

comment: The discriminant is $\Delta=-12$.

x^2+4:

1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984

comment: The discriminant is $\Delta=-16$.

x^2+5:

0.5282455737767852957397203365069928986112409518699295385412711022524927995386911843302490954454975139

comment: The discriminant is $\Delta=-20$.

x^2+6:

0.7130416261717549988168836394399263958626536556577069915726124806881331797368418897285270878184799893

comment: The discriminant is $\Delta=-24$.

x^2+7:

1.973043165919314116364213614985456888913242559104292733803020148161125168626087951745815308215853028

comment: The discriminant is $\Delta=-28$.

x^2+8:

0.7130631042401929902232626073998268375684482408392256661074012026495749206783764001179141688701402792

comment: The discriminant is $\Delta=-32$.

x^2+9:

0.9152089752121640060747951311513459121188887341705063719574894315518948178115801544851465379593919892

comment: The discriminant is $\Delta=-36$.

x^2+10:

1.082290322622870934008726464648342274750746063264927783711232919691927699363829647117420572025543324

comment: The discriminant is $\Delta=-40$.

x^2+11:

0.5101385751188618061860056763665242252404661892233736103082430377948040609666054252689766154191079408

comment: The discriminant is $\Delta=-44$.

x^2+12:

1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868

comment: The discriminant is $\Delta=-48$.

x^2+13:

1.285787607594973099821388187755306836651177012482711025958632819412813134808814396163949191153115206

comment: The discriminant is $\Delta=-52$.

x^2+14:

0.4203685120242415850612291867691435857669957869166042938990290458429773598292241811053479296717706134

comment: The discriminant is $\Delta=-56$.

x^2+15:

1.267019954790685681315350247382006105145768371868654178910125512950871437236030883609580774364585537

comment: The discriminant is $\Delta=-60$.

x^2+16:

1.372813462818246009112192696727018868178333101255759557936234147327842226717370231727719806939087984

comment: The discriminant is $\Delta=-64$.

x^2+17:

0.4917097430948072341233640382833226389861633259260834511662290709132051750511550610602398378623044735

comment: The discriminant is $\Delta=-68$.

x^2+18:

1.426126208480385980446525214799653675136896481678451332214802405299149841356752800235828337740280558

comment: The discriminant is $\Delta=-72$.

x^2+19:

0.9422204577474599205433576755296373799993034647265656299752433984881648313597146577902812948518917527

comment: The discriminant is $\Delta=-76$.

x^2+20:

0.5282455737767852957397203365069928986112409518699295385412711022524927995386911843302490954454975139

comment: The discriminant is $\Delta=-80$.

x^2+21:

0.6754967305699162440248975466908905318610594026999808150795609624059682995907108895254707935118439946

comment: The discriminant is $\Delta=-84$.

x^2+22:

1.766762995152103665725714568256978814731409229751309691306775260082484003131677025944646277449270985

comment: The discriminant is $\Delta=-88$.

x^2+23:

0.8166075041323586681283113818893528970340129178951570930204595471408835704259581029553830684300264873

comment: The discriminant is $\Delta=-92$.

x^2+24:

0.7130416261717549988168836394399263958626536556577069915726124806881331797368418897285270878184799893

comment: The discriminant is $\Delta=-96$.

x^2+25:

1.830417950424328012149590262302691824237777468341012743914978863103789635623160308970293075918783978

comment: The discriminant is $\Delta=-100$.

x^2+26:

0.3733514180011663008517026122804559993013766152391805932889147796919352213980823203194354170039387262

comment: The discriminant is $\Delta=-104$.

x^2+27:

1.120732753549291390618334186235437216763555081120662681776915461356405677172048151385740707615005868

comment: The discriminant is $\Delta=-108$.

x^2+28:

1.973043165919314116364213614985456888913242559104292733803020148161125168626087951745815308215853028

comment: The discriminant is $\Delta=-112$.

x^2+29:

0.4042724938761413201705112848081949197798504701269073648328608973002068055601973728845968281095300364

comment: The discriminant is $\Delta=-116$.

x^2+30:

0.8695751905453238912022643067607602377970713513389461248511546547987457009790152536448111667612905273

comment: The discriminant is $\Delta=-120$.

x^2+x+1:

2.241465507098582781236668372470874433527110162241325363553830922712811354344096302771481415230011736

comment: The discriminant is $\Delta=-3$.

x^2+x+3:

1.020277150237723612372011352733048450480932378446747220616486075589608121933210850537953230838215882

comment: The discriminant is $\Delta=-11$.

x^2+x+5:

1.884440915494919841086715351059274759998606929453131259950486796976329662719429315580562589703783505

comment: The discriminant is $\Delta=-19$.

x^2+x+7:

2.241465507098582781236668372470874433527110162241325363553830922712811354344096302771481415230011736

comment: The discriminant is $\Delta=-27$.

x^2+x+9:

0.9280103731273487855873097252803947565787508509905504594844853433869593978188827344697684992905128316

comment: The discriminant is $\Delta=-35$.

x^2+x+11:

3.259441847518388145093386059923914196725620715892583498750020991085655587697066611871551227021417497

comment: The discriminant is $\Delta=-43$.

x^2+x+13:

1.419857358966315738255882478053358602981063388340692803832065566567952327363225826438664468845011582

comment: The discriminant is $\Delta=-51$.

x^2+x+15:

0.7506330693483119384816695337893102809050716260116440015777356381958311886817350455720476745762888187

comment: The discriminant is $\Delta=-59$.

x^2+x+17:

4.174661613064506156384616027265554205258359532355780762832290173805074233803615278659632440732471028

comment: The discriminant is $\Delta=-67$.

x^2+x+19:

1.793172405678866224989334697976699546821688129793060290843064738170249083475277042217185132184009389

comment: The discriminant is $\Delta=-75$.

x^2+x+21:

0.9700198069727936513487153646506794009553303496568042714539174121546204557329979833218089012583478501

comment: The discriminant is $\Delta=-83$.

x^2+x+23:

2.141301828834744177405561293675416841044042531319356004932445116309036103158136828498402877268236243

comment: The discriminant is $\Delta=-91$.

x^2+x+25:

2.040554300475447224744022705466096900961864756893494441232972151179216243866421701075906461676431763

comment: The discriminant is $\Delta=-99$.

x^2+x+27:

1.143878070461191914460965968416389087615849533015381762745042519987791242580107196598246225550931869

comment: The discriminant is $\Delta=-107$.

x^2+x+29:

2.509924528638309698164020770571015618012214936531440762279479806568444102037061777052853261363088714

comment: The discriminant is $\Delta=-115$.

x^2+x+31:

2.482812754958659435877802042265728699767214193716973768198429923255571125959857518638268319299046271

comment: The discriminant is $\Delta=-123$.

x^2+x+33:

0.6596657567089936838732409315054822272005742886441638363420171151419848490461618510829897723956105335

comment: The discriminant is $\Delta=-131$.

x^2+x+35:

1.689602419947677057250855974880387107256813527569778238330400896888306148814092238563942505797773478

comment: The discriminant is $\Delta=-139$.

x^2+x+37:

2.689758608518299337484002046965049320232532194689590436264597107255373625212915563325777698276014083

comment: The discriminant is $\Delta=-147$.

x^2+x+39:

1.003074745695317297913893711663044683123966880363528175911436776511523502224541451536167360622096167

comment: The discriminant is $\Delta=-155$.

x^2+x+41:

6.639546354942843330647113715299775932937109171305971698307881455900526620852236299475102289608912434

comment: The discriminant is $\Delta=-163$; OEIS A221712 gives half of this normalization, and OEIS A331940 lists $41$ as a record addend [9].

Definition

For a monic irreducible quadratic polynomial $f\in\mathbb Z[x]$ whose values have no common prime divisor, the Bateman-Horn constant $C(f)$, also called the Hardy-Littlewood constant of $f$, is $C(f)=\prod_p\dfrac{1-N_f(p)/p}{1-1/p}$, where $N_f(p)$ is the number of roots of $f$ modulo $p$ [1] [3].

Parameters

$f$

— polynomial ($f=x^2+bx+c$ with $b,c\in\mathbb Z$, $f$ irreducible over $\mathbb Q$, and no prime divides every value $f(n)$)

Formulas

(1)

$C(f)=\prod_p\dfrac{1-N_f(p)/p}{1-1/p}$, where $N_f(p)$ is the number of roots of $f$ modulo $p$.

(2)

If $\Delta=b^2-4c$ for $f=x^2+bx+c$, then $N_f(p)=1+\left(\frac{\Delta}{p}\right)$, so $C(f)=\prod_p\left(1-\dfrac{1}{p-1}\left(\frac{\Delta}{p}\right)\right)$, with the Kronecker symbol $\left(\frac{\Delta}{p}\right)$.

(3)

With $\chi_\Delta(p)=\left(\frac{\Delta}{p}\right)$ and $L(1,\chi_\Delta)=\prod_p\left(1-\dfrac{\chi_\Delta(p)}{p}\right)^{-1}$, $C(f)=\dfrac{1}{L(1,\chi_\Delta)}\prod_p\left(1-\dfrac{\chi_\Delta(p)}{(p-1)(p-\chi_\Delta(p))}\right)$; the product in this formula converges absolutely.

(4)

The Bateman-Horn conjecture predicts $\#\{n\leq x:f(n)\text{ is prime}\}\sim \dfrac{C(f)}{2}\int_2^x\dfrac{dt}{\log t}$ for the quadratic polynomials in this table [1] [3].

Comments

(5)

The entries are $C(f)$ itself. Since every polynomial here has degree $2$, the Bateman-Horn asymptotic in (4) has leading constant $C(f)/2$.

(6)

For even $a$, every value of $x^2+x+a$ is even, so those polynomials are excluded; for odd $a$, no value is even, $N_f(2)=0$, and the factor at $p=2$ in (1) is $2$.

(7)

By (2), $C(f)$ depends on $f$ only through the character $\chi_\Delta(p)=\left(\frac{\Delta}{p}\right)$. Replacing $\Delta$ by $\Delta m^2$, where every prime divisor of $m$ already divides $\Delta$, leaves this character unchanged; $x^2+1$, $x^2+4$ and $x^2+16$ are one such group.

(8)

The Euler product defining $C(f)$ converges conditionally for these quadratic polynomials. The asymptotic interpretation as a density of prime values remains conjectural.

(9)

For $f=x^2+1$, OEIS A199401 gives $C(f)$ [5], while OEIS A331941 gives $C(f)/2$ [6]. For $f=x^2+x+41$, OEIS A221712 gives $C(f)/2$ [7], and OEIS A331940 lists $41$ as a record addend [9].

Programs

(P1)

PARI/GP

\\ Load https://oeis.org/A221712/a221712.gp.txt, which defines HardyLittlewood2.
default(realprecision, 140)
HardyLittlewood2(x^2+1)              \\ 1.3728134628182460091...
HardyLittlewood2(x^2+x+41)           \\ 6.6395463549428433306...

References

[1]

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation 16 (1962), no. 79, 363-367. (doi) (zbMATH) (MR)

[2]

Henri Cohen, High-precision computation of Hardy-Littlewood constants, 1998. https://oeis.org/A221712/a221712.pdf

Links

[3]

Wikipedia: Bateman-Horn conjecture

[4]

Wikipedia: Landau's problems

[5]

OEIS A199401: Hardy-Littlewood constant of $x^2+1$

[6]

OEIS A331941: half the Hardy-Littlewood constant of $x^2+1$

[7]

OEIS A221712: Hardy-Littlewood constant for $x^2+x+41$

[8]

OEIS A206709: number of primes of the form $b^2+1$ for $b\leq10^n$

[9]

OEIS A331940: record addends for Hardy-Littlewood constants of $x^2+x+k$

[10]

Karim Belabas and Henri Cohen: Computation of the Hardy-Littlewood constant for quadratic polynomials

Similar tables

Hardy-Littlewood singular series of prime tuples — linear Bateman-Horn systems give the Hardy-Littlewood singular series for prime tuples

Twin prime constant — the Bateman-Horn constant of the system $(x,x+2)$ is $2C_2$

Values of the prime zeta function at rational numbers — Cohen's acceleration rewrites the prime-indexed products through prime sums computed from zeta values

Values of Dirichlet L-functions at positive integers — (3) expresses these constants through Dirichlet $L$-values of quadratic characters

Residues of Dedekind zeta functions of quadratic fields — for a fundamental discriminant $D$, $L(1,\left(\frac{D}{\cdot}\right))$ is the residue at $s=1$ of the Dedekind zeta function of $\mathbb Q(\sqrt D)$

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds every polynomial $x^2+a$ with $-30\leq a\leq30$ satisfying the parameter constraint, and every polynomial $x^2+x+a$ with $1\leq a\leq41$ satisfying the parameter constraint)

How they were obtained:

Values were computed with the Belabas-Cohen PARI/GP algorithm [10], which implements Cohen's accelerated computation of Hardy-Littlewood constants [2]. PARI was run at $140$ decimal digits, and $100$ digits are written.

The root-count identity $N_f(p)=1+\left(\frac{\Delta}{p}\right)$ was checked by enumeration for every row and every prime $p\leq97$. The stored value of $C(x^2+1)$ agrees with OEIS A199401 [5] and with $2\times$ OEIS A331941 [6]; the stored value of $C(x^2+x+41)$ agrees with $2\times$ OEIS A221712 [7]. Recomputing $C(x^2+1)$ and $C(x^2+x+41)$ at $160$ decimal digits rounds to the same $100$ digits.